Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• It's a generalization of arithmetic, where variables represent numbers or quantities.
• Algebraic expressions, equations, and inequalities are fundamental building blocks.

Equations

• An equation is a statement that asserts the equality of two expressions.
• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• These values are called solutions or roots of the equation.
• Linear equations involve variables raised to the first power.
• Quadratic equations involve variables raised to the second power.
• Systems of equations involve two or more equations with the same variables.

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• A polynomial of degree n has the general form: a_n*x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where a_n ≠ 0.
• The degree of a polynomial is the highest power of the variable.
• Polynomials can be added, subtracted, multiplied, and divided.
• Factoring a polynomial involves expressing it as a product of simpler polynomials.
• The roots of a polynomial are the values of the variable that make the polynomial equal to zero.

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• The input is called the argument of the function, and the output is called the value of the function.
• Functions are commonly denoted by symbols such as f, g, and h.
• The domain of a function is the set of all possible inputs.
• The range of a function is the set of all possible outputs.
• Functions can be represented graphically, algebraically, or numerically.
• Common types of functions include linear, quadratic, exponential, logarithmic, and trigonometric functions.
• Function composition involves applying one function to the result of another.

Matrices

• A matrix is a rectangular array of numbers arranged in rows and columns.
• Matrices are used to represent linear transformations, solve systems of equations, and store data.
• The dimensions of a matrix are given by the number of rows and columns (e.g., an m x n matrix has m rows and n columns).
• Matrices can be added, subtracted, and multiplied.
• Matrix multiplication is not commutative in general.
• The determinant of a square matrix is a scalar value that provides information about the matrix's properties.
• The inverse of a square matrix (if it exists) is a matrix that, when multiplied by the original matrix, yields the identity matrix.
• Transpose of a matrix is obtained by interchanging its rows and columns.
• Eigenvalues and eigenvectors are important concepts in linear algebra with applications in various fields.

Complex Numbers

• A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
• The real part of a complex number a + bi is a, and the imaginary part is b.
• Complex numbers can be represented graphically on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
• Complex numbers can be added, subtracted, multiplied, and divided.
• The conjugate of a complex number a + bi is a - bi.
• The magnitude (or modulus) of a complex number a + bi is the distance from the origin to the point representing the complex number on the complex plane, given by √(a^2 + b^2).
• Euler's formula relates complex exponentials to trigonometric functions: e^(ix) = cos(x) + i*sin(x).
• Complex numbers are used in various fields, including electrical engineering, quantum mechanics, and fluid dynamics.
• De Moivre's theorem states that (cos(x) + isin(x))^n = cos(nx) + isin(nx).